In calculation, the degree was conceived to be divided into 60 _minutes_, the minute into 60 _seconds_, and so on. But in practice it was impossible to divide the limb of the instrument into parts so small. The armils of Alexandria were divided into no parts smaller than sixths of degrees, or divisions of 10 minutes. The angles, observed by means of these divisions, were expressed as a fraction of the circumference. Thus Eratosthenes stated the interval between the tropics to be 11/83 of the circumference.[85\3] [Note 85\3: Delambre, _A. A._ i. 87. It is probable that his observation gave him 47⅔ degrees. The fraction 47⅔/360 = 143/1080 = 11 ∙ 13/1080 = 11/(83+1/13), which is very nearly 11/83.] It was soon remarked that the whole circumference of the circle {164} was not wanted for such observations. Ptolemy[86\3] says that he found it more convenient to observe altitudes by means of a square flat piece of stone or wood, with a _quadrant_ of a circle described on one of its flat faces, about a centre near one of the angles. A peg was placed at the centre, and one of the extreme radii of the quadrant being perpendicular to the horizon, the elevation of the sun above the horizon was determined by observing the point of the arc of the quadrant on which the shadow of the peg fell. [Note 86\3: _Synt._ i. 1.] As the necessity of accuracy in the observations was more and more felt, various adjustments of such instruments were practised. The instruments were placed in the meridian by means of a _meridian line_ drawn by astronomical methods on the floor on which they stood. The plane of the instrument was made vertical by means of a plumb-line: the bounding radius, from which angles were measured, was also adjusted by the _plumb-line_.[87\3] [Note 87\3: The curvature of the plane of the circle, by warping, was noticed. Ptol. iii. 2. p. 155, observes that his equatorial circle was illuminated on the hollow side twice in the same day. (He did not know that this might arise from refraction.)] In this manner, the places of the sun and of the moon could be observed by means of the shadows which they cast. In order to observe the stars,[88\3] the observer looked along the face of the circle of the armil, so as to see its two edges apparently brought together, and the star apparently touching them.[89\3] [Note 88\3: Delamb. _A. A._ i. 185.] [Note 89\3: Ptol. _Synt._ i. 1. Ὥσπερ κεκολλήμενος ἀμφοτέραις αὐτῶν ταῖς ἐπιφανείαις ὁ ἀστὴρ ἐν τῷ δι' αὐτῶν ἐπιπέδῳ διοπτεύηται.]